Theorem ralpr 2432

Description: Convert a quantification over a pair to a conjunction.

Hypotheses

Ref	Expression
ralpr.1	⊢ A ∈ V
ralpr.2	⊢ B ∈ V

Assertion

Ref	Expression
ralpr	⊢ (∀x ∈ {A, B}φ ↔ ([A / x]φ ⋀ [B / x]φ))

Distinct variable groups:   x,A   x,B

Proof of Theorem ralpr

Step	Hyp	Ref	Expression
1		df-ral 1652	. 2 ⊢ (∀x ∈ {A, B}φ ↔ ∀x(x ∈ {A, B} → φ))
2		visset 1816	. . . . . 6 ⊢ x ∈ V
3	2	elpr 2428	. . . . 5 ⊢ (x ∈ {A, B} ↔ (x = A ⋁ x = B))
4	3	imbi1i 186	. . . 4 ⊢ ((x ∈ {A, B} → φ) ↔ ((x = A ⋁ x = B) → φ))
5		jaob 424	. . . 4 ⊢ (((x = A ⋁ x = B) → φ) ↔ ((x = A → φ) ⋀ (x = B → φ)))
6	4, 5	bitr 173	. . 3 ⊢ ((x ∈ {A, B} → φ) ↔ ((x = A → φ) ⋀ (x = B → φ)))
7	6	albii 1001	. 2 ⊢ (∀x(x ∈ {A, B} → φ) ↔ ∀x((x = A → φ) ⋀ (x = B → φ)))
8		19.26 1069	. . 3 ⊢ (∀x((x = A → φ) ⋀ (x = B → φ)) ↔ (∀x(x = A → φ) ⋀ ∀x(x = B → φ)))
9		ralpr.1	. . . . 5 ⊢ A ∈ V
10	9	sbc6 1960	. . . 4 ⊢ ([A / x]φ ↔ ∀x(x = A → φ))
11		ralpr.2	. . . . 5 ⊢ B ∈ V
12	11	sbc6 1960	. . . 4 ⊢ ([B / x]φ ↔ ∀x(x = B → φ))
13	10, 12	anbi12i 484	. . 3 ⊢ (([A / x]φ ⋀ [B / x]φ) ↔ (∀x(x = A → φ) ⋀ ∀x(x = B → φ)))
14	8, 13	bitr4 176	. 2 ⊢ (∀x((x = A → φ) ⋀ (x = B → φ)) ↔ ([A / x]φ ⋀ [B / x]φ))
15	1, 7, 14	3bitr 177	1 ⊢ (∀x ∈ {A, B}φ ↔ ([A / x]φ ⋀ [B / x]φ))

Syntax hints:   → wi 3   ↔ wb 146   ⋁ wo 222   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  [wsbc 1172  ∀wral 1648  Vcvv 1814  {cpr 2414

This theorem is referenced by:  rexpr 2433

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-sbc 1945  df-un 2053  df-sn 2416  df-pr 2417

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